How to make Apache serve index.php instead of index.html?

As others have noted, most likely you don't have .html set up to handle php code.

Having said that, if all you're doing is using index.html to include index.php, your question should probably be 'how do I use index.php as index document?

In which case, for Apache (httpd.conf), search for DirectoryIndex and replace the line with this (will only work if you have dir_module enabled, but that's default on most installs):

DirectoryIndex index.php

If you use other directory indexes, list them in order of preference i.e.

DirectoryIndex index.php index.phtml index.html index.htm

As of today (2015, Aug., 1st), Apache2 in Debian Jessie, you need to edit:

root@host:/etc/apache2/mods-enabled$ vi dir.conf 

And change the order of that line, bringing index.php to the first position:

DirectoryIndex index.php index.html index.cgi index.pl index.xhtml index.htm